We get better by practicing only because of our brain's neuroplasticity, which one might argue, is a mechanism of adaptation, and thus a result of evolution. An intelligent life might have developed other ways of coping with change, and not necessarily through a neuroplastic brain, but for example, through an instant rewiring of the brain that doesn't require any practice.

The same can be said for justice, even though it might seem as a social construct, it's still rooted in our biology (emotions are still "physical" reactions in our body) and so an intelligent life might have developed a different biological system different than that of emotions.

One theory is that some intelligences evolved the opposite to us. They started in pure silicon form and developed super-intelligence in the early stages and didn't evolve from carbon. They were /born/ computers and evolved from there into hyper-intelligence capable of exploring galaxies with Von Neumann probes[0]

[0] https://futurism.com/von-neumann-probe

Also, even though silicon has interesting properties on its own, carbon is kind of just better at everything else. It can make bigger and more stable structures, for example. Its oxide is also not, you know, a stable and unreactive solid. So it would be surprising for life forms not to use carbon extensively, unless it was much too rare, and in that case I honestly doubt it would get very far. As a building block, carbon is just outstanding, there is a reason life on Earth is based on it even though there is a thousand times more silicon than carbon on the planet.

You might have an alien species that can copy the brain state of someone who already knows a task, or one that can gain skills extremely quickly through practice, but I don't think practice is going away. The fact that neural networks have been the only way we've been able to solve a lot of problems, which approximate the way that the human brain learns, is pretty strong evidence of this. You start out with some weights, you measure loss, you adjust weights, and then you try again.

Justice as an emotion might go away or exist in a different form, but the underlying reason why humans have a sense of justice is evolutionary psychology / game theory. Probably any life form which is shaped by evolutionary forces would have some similar instinct. (Certainly not all possible intelligent life forms though, I'd agree.)

Chances are a lot of machine learning implementations can be replaced with just "boring" statistical models and achieve more power.

OTOH plenty of creatures don't need to learn. Does a mosquito need to learn? No, it spawns thousands of offspring and doesn't live very long. The high spawn rate means you have a wide variety of natural mutations in your offspring, meaning one or a few of them are likely to have higher fitness in a given niche. It doesn't matter if most die if a few go on to survive. This is the strategy many organisms use to dominate the world in far greater numbers than our own species.

This is of course true, but the innovation in ML is not that a neural network (or whatever model) is equivalent to something else, it's finding the weights in the first place.

These are all difficult assertions to make because we're trying to prove a negative. Your claim is evidence that neural networks are one way to do it, and nobody's arging that. But it's not evidence that there doesn't exist a better way, or that alien life might evolve a different and even less optimal way.

Legs are pretty ubiquitous, even flying insects have them. If we'd never seen a fish or snake we might conclude that they're inevitable. Somebody that evolved to be rad-hard on a planet without a magnetosphere might conclude that life can't exist without the thick carapace that they're made of and only look for planets rich in silicon and calcium.

We know that evolution has limitations in how it explores the state space; sometimes certain new developments depend upon past developments. But it seems to me that the development of a brain would hit relatively few of these barriers.

How does the rewiring system know what the ideal end state is in advance, and how does the alien evaluate if the new state is fit for purpose?

I'm with you on justice though. A parasitic species might have a completely different view of rights and obligations than we do.

(2) The question of "functionally equivalent" is interesting here. If an alien species accurately predicts the trajectory of a complex rocket, many of us would say they must be using math, because we know that our mathematics governs how rockets move. Even if they cannot communicate their method in a way we can understand. Economists sometimes use the phrase "as if", e.g. the alien acts "as if" they have a utility function, regardless of whether that's how they conceptualize their own action. Similarly. they would certainly act "as if" they had math and physics. But the question of whether that would imply they "do", I'm less sure of.

Given that even people (loosely) in the same culture often disagree about what constitutes "justice" and use the term in mutually exclusive ways, we should definitely bet against the proposition that "all intelligent beings" understand it.

Justice - rendering to each person what is owed to them.

It's obvious that we will very often disagree about 'what is owed', but doesn't our passionate disagreement in this case show that (1) we agree that practicing justice is good and (2) we are closely aligned on the existence of this thing called 'justice'?

For instance, in climate justice we might say that we owe something to all of the people who are young now or haven't been born yet. In social justice you might owe something to a whole group of people. Just because they're groups doesn't mean the individual people that comprise the group dissolve into an abstract concept, even if that's how it might seem in our minds.

In some cases maybe the definition is too narrow because it depends on what is or isn't a "person" which we might define too narrowly. Do we owe things to animals? I'd say so, and I think most people would agree, depending on the animal. Do we owe things to plants? Maybe. Do we owe things to the planet Mars, assuming it's entirely devoid of life? I don't know, but I think it would offend people's sensibilities if we were to dump toxic waste all over it's surface, even if we were sure that humans are never going to settle there or use any of its resources. There's a view of climate justice that we owe things to the Earth directly rather than (or in addition to) owing something to the people that live there or will live there.

I shared an argument above for why it can be viewed as a universal and judging by your comment above you are somewhat skeptical of this claim.

If we shift the discussion to allow conceptions of 'justice' that move away from the classical tradition and include modern ideas like 'climate' justice or 'social' justice, I will revert to agreeing with your skepticism.

I don't think anyone can plausibly claim that these more marxist-oriented modern definitions are universals.

This is extremely subjective.

Justice does not have to be cogent, which is defined as "(of an argument or case) clear, logical, and convincing.". There were and are justice systems that leave out one or more of these ingredients to some extent. Some leave out some logic by presuming the existence of a supernatural being. Some are more authoritarian and not very convincing.

What do you mean by that?

So does the definition of justice.

> vigilante justice

Except 'vigilante justice' and 'justice' are concepts that have little to do with each other. You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.

That depends on how well different people's definitions of justice line up. There are many things that vigilantes can enforce pretty well.

> You might as well be talking about the 'de facto law' (like for instance the "law" enforced by the local mafia) vs. the 'de jure law' here.

Sure, why not? Mafia law is often not justice, but I think it qualifies as law where sufficiently powerful. You seem to think this argument debunks itself?

If reputation and peer pressure scaled to group sizes bigger than Dunbar's number (i.e. about a hundred or so) then we probably wouldn't need laws at all.

I suppose you one might say that social expectations are just another kind of law, in which case, yeah it's hard to imagine any group of people without some kind of expectations of how each other will behave. That's kind of the basis of human relationships.

* A society of one.

* A society of one family isolated in nature, where each member is allowed to express their peculiarities and eccentricities, but never do each other any harm - not because of established rules, but because they truly love and care of each other.

The second example is virtually unreal (and even expressions of 'love' and 'care' can be harmful).

[0] https://en.wikipedia.org/wiki/Prisoner%27s_dilemma#The_itera...

Hey, we can always claim that those who disagree with us are not intelligent.

I'm only partially joking. Lots of today's "justices" have so many internal contradictions that I feel like we should separate them into their own category.

This might be better than mathematical constructs which conceivably do not require technology, "just" thought to discover.

The problem here is not even in the need to spend colossal amounts of energy to signal one's presence. The question is whether such advertising is a good idea if the civilization strives to survive; very likely it is not.

There is no reason to believe the symbolic encoding would be the same. For example, if they evolved with 8 fingers instead of 10 they could be driven to encode their symbols in a base 8 system.

I think the point of the OP sharing the PBS spacetime video in this context is that we have reason to believe that the fine structure constant is truly constant throughout the universe. So if the aliens have the capability of measuring and reasoning about electron orbitals and their binding properties with nucleus, then there will likely be significant overlap between our mutual concepts of 'fine-structured-constantness'. Just made that word up. :-)

This would be independent of encoding scheme and independent of practical communication issues.

It's an interesting thought exercise but I don't think there is any reason to expect mathematics to be the same across all potential life in the universe. Human mathematics is adapted to human evolutionary contingencies and the same would be true for alien mathematics, physics, and engineering. By definition of "alien" their mathematics would be alien to us and even if there were commonalities they would be very hard to uncover.

I'm comfortable living with the tension between these two propositions: (1) The manifestation of laws of physics are real, measurable, spread across the galaxies, and essentially outside of our subjective experience. For example, I do believe that gravity and electrons are real ... not just real for me. (2) The way in which we interpret these physical realities is somewhat conditioned by the constraints and experience of being messy human animals.

You can dump energy into two frequencies. You're overcomplicating it.

The video suggests binary which is also pretty simple.

PG's essay seems to (implicitly, not explicitly) compare between technologically-sophisticated intelligence - ie not between octopuses and humans, but between species that could at least communicate through inter-stellar distances, if not traverse them directly. If so, convergent evolution would have imposed even more onerous similarities between such species: to develop a radio-transmitter, intelligent creatures would also likely to have been very socially developed, or how else would they have been able to transform the resources around them to build a transmitter?

Therefore it is the physical world in which we live that most likely leads us to conceive "math" similarly... and even "justice." [1]

[0] https://www.quantamagazine.org/arik-kershenbaum-on-why-alien...

[1] tit-for-tat being one of the best strategies to solve an infinitely-recurring prisoner's dilemma

I am reading the sci-fi book The Sea of Rust right now, that takes place on earth after AI’s have finally killed the last humans. In this fictional work, AIs take on human traits and I don’t find that “believable” even for sci-fi.

For example, even if silicon-based AI replaced humans on earth, it would eventually [0] need to find a way to power itself / continue itself. Ultimately, it would have to revert to solving the same "problem of life"... how do we transform energy/entropy available in the environment into something that "perpetuates the system." When that happens, this AI will itself become subject to the forces of natural-selection, and - over a long enough period of time - naturally-selected traits will be re-aquired (even if such traits were "lost" during a human-to-AI hand-over).

[0] Yes, of course, there could be a very large period of time during which currently constructed energy infrastructure continues on... and this period could be measured in hundreds/thousands of years... very long in terms of human lifespans, but not in geological terms.

There is overlap between pg's ideas and what in the classical tradition is called Natural Law Theory. PG may or may not be interested in drawing out the connection, but since he references Aristotle I have to believe he is at least aware of a touch point.

To give a distilled definition, Natural Law Theory is the application of the laws of nature to rational creatures.

In the context in which NLT developed, the only free rational creature was the human being. But both AI development, and concepts from evolution through natural selection, potentially allow us to apply aspects of the theory to different rational agents.

I chuckled at this one because one of the first things I was taught early in science class was that the root of "to invent" is latin "invenire", which translates to "to find", "to discover".

Kind of a humbling moment to realise we're not creating anything, merely too blind to see, groping in the dark for truths that lay out there for us to trip upon.

These are pretty strong statements for which there’s no arguments provided for but serve as assumptions for the rest of the article. I don’t think there’s consensus among mathematicians, philosophers, cognitive scientist, or biologists on this.

Mathematics is most definitely a human endeavor, and so we can’t really make claims about its existence in the universe independent of humans. I think alien analogs to mathematics are unlikely to match ours. If we are lucky, I think it could be the case that the various structures could be similar, but the likelihood the implementations resemble each other are slim. It’s even a stretch to assume the structures would relate. Even humans do not fully agree on mathematics. There is no “one” mathematics because mathematics is the human exploration of idealized objects using a variety of human logical systems.

And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand.

I think the idea is that math is not created by humans, but documented by humans. Sure, the specific terminology may be our invention, but there are basic mathematical properties that seem (from our perspective) like they should be universal. For example, whatever names a being has for the numbers 1 and 2, if you take that 1 and add 1 more, you must get 2 (or the "local equivalent") as the result.

My guess is that, if what we call math isn't truly universal, it's probably at least universally true within the realm of physical life, and there's likely some massive causal chain from the root properties of physics itself to the mathematical properties that we call "math". When it comes to raw, untethered "consciousness" (or whatever one would prefer to call it), this may not hold true even in the slightest.

Yes, this comment steps slightly outside what could ever be determined purely by the scientific method at the end. I feel it is useful to do so in discussion, even when that cannot directly enter into research. There are some truths to the larger universe that I don't think the scientific method will ever truly be able to uncover, just due to it's rigor. Some aspects of the universe are just simply not falsifiable, but they're still worthy of discussion with an open mind.

Responding to your second point, I'm afraid I can't agree with you that unfalsifiable propositions are "useful" discussion contributions - especially not with an "open mind". The only criterion on which such propositions can be judged is whether they are fun to believe, and that is a very dangerous muscle to flex.

So aliens won't be able to count? They won't have a concept of zero? They won't have a concept of 1=successor(0)? I find this very, very hard to believe, and a lot of mathematics follows from the structure of the natural numbers.

If you accept evolution by natural selection is a universal law, then I think it naturally follows that ability to count must evolve. After all, it's pretty important to know whether there are 0, 1, or many predators/food/prey/enemies.

Sorry. That's the one. That's the one that broke me. Jeremy Bearimy

I honestly can't imagine how you can reach this conclusion with any rigour. Do you agree that aliens will need to consume some energy source to stay alive, which we will call "food"? Do you agree that an understanding of "there's no food in my environment", "there's some food in my environment", and "there's lots of food in my environment" would be selected for? I certainly hope so, so at the very least they will understand the differences between zero, non-zero and "many".

The only way this wouldn't happen is if the environment is so rich in abundance that there is never any absence of food. But this is impossible, because even single-celled life by necessity will reproduce to consume all available resources until it reaches an equilibrium matching the rate of food production. So any intelligent species will necessarily evolve in an environment of scarcity where zero and non-zero will be implicitly understood.

Since intelligent life will necessarily evolve in scarcity, quantifying the amount of food is a useful trait that would be selected for. This is why we've now proven that numerous "non-intelligent" animals can count, including salamanders, chicks, mosquitofish, honeybees and more. Intelligent life needs to understand where they are, what they have and what they will need in the future. This involves quantifying, aka counting, no way to escape it.

> There's a vast gap between counting and what mathematics is and encapsulates.

Yes, but you posited intelligent aliens that have their own math. The conclusion that they would not understand zero and repeated application of a construction over zero to build non-zero quantities is impossible. It is the very root of building a theoretical structure of any kind, so if they have math of any kind, they have some kind of counting system that will have an isomorphism to ours.

And you believe that aliens that can count is something that is "arguable".

You're the dot.

There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths.

The math that we invented is influenced by our biology and capacity to sense our environment. Our brains and how those brains work with our sense organs. This pattern is likely universal (all life will have methods of sensing their environment and interacting with it), but the methods might be very different.

0 and 1 are both valid probabilities.

>What about an organism that can see/focus/sense multiple things simultaneously, and a single "thing" is a set.

Its possible, using sets only containing other sets (or possibly the empty set), to construct the integers.

>There could be an entirely different paradigm to "counting" and consequently to the fundamentals of maths.

Systems of mathematical expressions are just like coding languages. The choice is arbitrary, one can always emulate the job of another. Just like how I did with your chosen examples, in principle one can always hack the integers out of whatever system you give me (or hack whatever system out if integers).

If not, then you have to explain how an alien species might develop that is not subject to physical constraints and evolution by natural selection.

If so, then you must agree that any alien must be able to distinguish two scenarios, "I sense some food here" and "I sense no food here". The basic binary distinction is inescapable, and this is the foundation of true/false, 0 and 1, etc.

Rather it is the concept of objects remaining in a single place that would require some real mathematical innovation to a creature with no experience of such an idea. And so this distinction of entirely separate logical states, far from being basic or inescapable, is our very human invention. It is useful for creatures like us, who perceive things in one place when they are not really so, who do their computing with sand in a region where it's bountiful, and who encode abstractions as software because doing so in dedicated hardware is more costly.

While it is certainly possible that all intelligent life would have these constraints, there is no particular reason to expect it. What we can expect is that humans will expect others to be too much like ourselves; it's a well-known cognitive defect in our species.

Plants don't just subsist on photons, there are many other ingredients.

> We know from heseinberg that the sensing of this food "here" or "there" is nonphysical.

I don't know what this means. How do you "non-physically" sense photons?

> Rather it is the concept of objects remaining in a single place that would require some real mathematical innovation to a creature with no experience of such an idea. And so this distinction of entirely separate logical states, far from being basic or inescapable, is our very human invention.

Assuming you're talking about some alien made of bosons that aren't subject to the Pauli exclusion principle, you'll note that bosons still interact with fermions in which that principle does apply, so I don't think your argument follows. I admit I don't really understand your premises though so I have no idea what you really meant.

you said every thing which implies discreteness. Also, probabilities are probabilities of an event (read: something discrete) happening

There are intelligent beings on Earth that don’t seem to even have analogs to human mathematics, at least that are apparent to us. We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical of claims that alien thinking will bear any resemblance to human thinking.

Yes and no. You don't need more structure than 0 and 1 to describe literally any form of information, and we're using machines right now that use such an encoding. The idea that any organism of sufficient complexity to have any kind of math won't have any notion of 0 and 1 is very implausible.

That said, we certainly won't have the same syntactic descriptions of most structures, but they will certainly be relatable via isomorphisms.

> We can barely communicate with a small subset of animals and plants on Earth. So I am just inherently skeptical that claims that alien thinking will bear any resemblance to human thinking.

But what does that have to do with math? Math isn't about how thinking works, it's about how structures are related to each other. Structures and their relations don't depend on how one thinks. As above, how such structures are described/encoded probably depends on how one thinks (aliens maybe won't use pencil and paper), but the structure being described will be the same and so there will necessarily exist some kind of isomorphism between their "syntax" and ours, as syntax is a projection of the structure.

Even plants have observable behaviour showing a distinction between 0 and 1: they observably move towards the sun when it's shining, and don't move when it's not. This isn't knowledge of "math", but simply to demonstrate that structure is everywhere and life simply must develop some intrinsic understanding of it.

Mathematics is also shaped by our thinking, which was my point. I think it’s a strong claim that aliens would even have a “mathematics”.

I mean "no" to your implicit assertion that such basic logical and axiomatic systems would not evolve in any alien species capable of mathematics. Any such alien will distinguish true and false, will have AND, OR and NOT connectives, and will understand a form of implication (it's inherent to causality). That's all you need to build an understanding of most of our formal systems.

Yes the particular expression of our information theory and computer science depends on specific syntactic choices which implies a surface dissimilarity, but the underlying structure will be the same even when expressed in alien math.

For instance, an alien species might evolve in an environment in which hyperbolic geometry is more natural (say a species large enough that they can sense gravity directly), and so they develop that geometry first. This will have an isomorphism to our formal model of hyperbolic geometry, and we can then explain Euclidean geometry to them from there.

Edit:

> Mathematics is also shaped by our thinking, which was my point.

Yes, but ultimately irrelevant. This drives the pace of mathematical discovery, and what kinds of mathematical formulae we develop or find most interesting, but this is ultimately irrelevant to the fundamentals which underpin all math, which is what this really comes down to.

Implication doesn't have anything to do with causality and in fact the concept of implication in mathematical logic is broken. See: the paradoxes of material implication:

https://en.wikipedia.org/wiki/Paradoxes_of_material_implicat...

To simplify, F -> T (true if false) is a true implication so, for example, I can say that "I am the pope therefore it rained yesterday" and, if it rained yesterday, then the implication is true even though I am not the pope. There has been endless grumbling among philosophers and mathematicians because of this kind of paradox but it is an inevitable result of the axiomatic definition of implication by means of a truth table, and there's no way to correct it without also changing the truth tables of disjunction and negation (because A OR NOT B is equivalent to NOT B THEREFORE A, i.e. because of the way disjunction and negation work, false implies true; you will have to work through this on your own and hit your head on your desk very hard, many times, just as I did when I first realised what a mess this is).

In other words, either we accept human axioms of logic, and we have paradoxes of implication, or we don't have paradoxes of implication but then we don't accept human axioms of logic. An alien civilisation may well choose to not accept any axioms of logic that lead to paradoxes of material implication, so they won't have human axioms of logic and, if their formal system is sound, they won't have human logic, and therefore, no human mathematics.

In other words, no, aliens will not necessarily have the same mathematics as humans.

I disagree, it's seems very obvious that if-then connectives are a crude causal description. Yes, the crude form is problematic because it's crude.

All the things you see around you are an outcome your brain processing. That applies to any structures that you abstract from that as well. Math is an exploration of how the brain does that.

Is a shapeless creature even logically coherent? Intelligence needed for math requires making distinctions, and distinctions imply structure, and structure is logically incompatible with true "shapelessness".

Do you see how your argument is self-defeating?

According to logic that humans have developed, there is such thing as a "shape". But Western philosophers have pondered the innateness of a "shape" or an "object" from very early on (Plato, through Leibniz, beyond).

"Shape" and "logic" are both human constructs articulating "structure", another human construct.

A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence; logic, truth, and structure are features that have emerged from human intelligence. I wouldn't accept the argument that an entity must exhibit the same features to qualify as intelligent simply because humans have.

There is such a thing as "structure", of which "shape" is an instance, yes.

> "Shape" and "logic" are both human constructs articulating "structure", another human construct.

Structure is not a human concept. We have particular conceptions of structure, but structure exists, period. 0 != 1, they have different structure. This is indisputable.

> A shapeless creature doesn't need to be "logically coherent" to exhibit intelligence

If you think that reality does not have to be logically coherent, or that that does not necessarily imply that any creatures within reality have to have a logically coherent description consistent with coherent natural laws, then you're talking about a fantasy world of your imagination and I don't think there's anything further to discuss.

Most of the world did mathematics for a long time without zero (I hope you know that most number systems like Roman didn't have zero till that eventually came from India, and we evolved to have the current number system). Who knows what direction different number systems might have taken if they didn't come in contact with zero.

Firstly, I disagree that humanity managed without zero. Literally everyone had an intuitive understanding of zero, they just didn't have it in their formal systems that were being studied by philosophers. For instance, try walking walking up to a vendor in Ancient Greece and just taking something without paying.

Secondly, it's largely irrelevant because a lot of math with zero can be mapped to math without zero with no loss of information, so even if aliens used math without zero there would be no trouble communicating as there would still be an understandable formal correspondence.

I also don't think any alien species with which we will communicate will not understand zero. It just seems impossible. Before philosophers came up with zero in formal models, everyone intuitively understood the concept. Every animals knows when they have no food vs. when they have some food. Humans in ancient civilizations also couldn't just take something without paying.

Almost certainly not, but they're probably isomorphic. And either way if we show them our axioms they will be able to validate our mathematics and vice versa.

The truths of mathematics are of the form 'if A then B'. Even if they don't start at A or even accept A as true, but will should still get B if they assume A.

That's a strong statement. I'd probably talk about homomorphism.

Maybe not in its entirety, but I find it hard to imagine that any civilization as advanced as ours (let's say a civilization that manages to harness nuclear fission, just to set a baseline) will not come up with concepts such as prime numbers, real number, complex numbers, calculus, etc. If they do that, they will inevitably find the same structures we found using function theory. They will know about differential equations and prove similar theorems about them as we did. And the Pythagorean theorem is a universal truth that holds everywhere.

Or at least in every world where there exist straight lines, yes? For instance:

> The Pythagorean theorem is derived from the axioms of Euclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theorem implies, and is implied by, Euclid's Parallel (Fifth) Postulate.[59][60] Thus, right triangles in a non-Euclidean geometry[61] do not satisfy the Pythagorean theorem. For example, in spherical geometry, all three sides of the right triangle (say a, b, and c) bounding an octant of the unit sphere have length equal to π/2, and all its angles are right angles, which violates the Pythagorean theorem because {\displaystyle a^{2}+b^{2}=2c^{2}>c^{2}}.

https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclid...

Some humans believe that the prominence of real numbers is a historical accident. It seems quite plausible to me that a human society, much less an alien one, would go down a mathematical evolutionary path based on the constructable numbers and the computable numbers.

Regardless of whether we eventually find the same structures, there are things that we might consider basic which they find esoteric and vice-versa.

Heaven forbid we encounter an alien civilization that discovered an O(log n) algorithm for integer factorization before they invented steam power.

That is a great point. The continuum is contentious and pretty highly debated philosophically.

I also encourage people interested in that to also learn about smooth infinitesimal analysis. Just a small tweak in the underlying logic and model yields unique mathematics and questions established assumptions.

Is there a “proof” of that? How could a proof exist? You’d probably win several awards if you had one.

> It is just a collection of arbitrary abstract definitions and what follows from them.

… created by humans.

It's how we define the concept "mathematics". If a result was dependant on "biology or perception of reality" or anything else outside its defining axioms, it wouldn't be mathematics.

A book I might recommend and that I'm going through at the moment is Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Rafael Nunez. The origin and meaning of mathematics is strongly influenced by cognitive sciene, and thus biology. I've been downvoted, but this is not a totally novel or off the rails idea. It is basically accepted in robotics that embodied cognition is how you get a robot to understand and perceive its environment. Where do you think that idea came from?

I haven't read that book, but I did just read the wikipedia page for it, for what it's worth. Based on that I feel it's more that they're arguing that the path we've taken and the metaphors we've used while exploring our way to our current understanding of mathematics is based on our biology. Our biology has greatly affected the order we've discovered things and how we understand those thing and how we actually 'do' math day to day, and all that I agree with. It's also very likely that aliens will have taken a very different path and have very different metaphors and proofs for understanding and doing their version of what we call mathematics. Because of this it might very well be very difficult for us to initially understand each others mathematics.

Indeed looking through human history our philosophical understanding of mathematics fundamentally 'is' has changed many times. Yet mathematical truth's we've found along the way have always remained constant (barring errors in calculation or reasoning) even as our understanding of mathematics has changed.

I believe that once we gotten passed all that both us and the aliens will find that, at the core, we both agree on what is "mathematically" true.

I will however also concede that some of the arguments in this thread has made me slightly less sure than I was before, so that is something I guess.

Even the concept of "number" itself is almost certainly an artifact of perception that enabled our ancestors to survive our niche, planetary environment, and not an inherent feature of objective reality.

Aliens, evolved to survive in another environment entirely with a different set of initial conditions, almost certainly would not have the same, nor even any, understanding of "number".

Would we consider such aliens a civilization or some kind of insensate "process"?

I'm so over this deconstructionist "reality is your perception" line.

Numbers are a thing. In fact they're one of the most basic observable things about the universe. And the concept if a number holds up all the way down to the quantum level. I.e. space and time are discrete and therefore space and time can both be COUNTED. Counting is literally one of the most basic and early achievements of human cognition and were gonna act like we just made it up?

Absolute silliness.

What makes you completely reject that? I don't think it has as much to do with deconstructionism as it does embodied cognition. We keep learning more and more about how biology and physiology and evolutionary pressures affect and inform cognition and thus perception. I was recently reading about how there have been scientific studies that seem to suggest that certain animals seem to experience time differently that we do. So you could say "time is a thing", but yet, it appears that it is not the same thing across lifeforms. There are animals that sense gravitational and electromagnetic fields, something we cannot do. Would it make sense to them to say "all beings can read these fields because we do"?

I think the problem is that it is all too easy to fall into the trap in thinking that alien lifeforms would be like us. There's a multitude of evidence of that here on Earth in the variety of life, despite even coming from the same origin.

That actually proves the fact that reality is not simply someone's perception. (We humans do not perceive these fields, and so it took scientific advances for us to discover them as part of the objective reality.)

There is an objective reality that exists beyond our perceptions. But our perceptions are based on that objective reality to some extent. We're not just making shit up. That doesn't make any sense.

You imagined what I meant so vividly that you literally made up a quote!

Try responding to what I actually wrote, and if it's unclear, asking with some humility.

I'm not even sure what your objection is, exactly. Nothing you wrote contradicts what I wrote.

Kant would have a field day with this statement

https://cosmosmagazine.com/technology/artificial-intelligenc...

Different truths that describe the same system.

https://cosmosmagazine.com/technology/artificial-intelligenc...

In any case, any model which includes infinity (and Peano arithmetic already does) is pure convention and unconsciously assumes a lot of things.

Finitary induction may make sense as something "universal". Further than that, we are making things up as we go (and I am a professional mathematician). The fact that they work to solve real problems does not make them more real.

It is a tautology that the famous incompleteness results in mathematics and computer science are the result of the axioms of arithmetic used to derive them. Would Gödel be able to derive his incompleteness result without Peano's axiomatisation of arithmetic? Not really. Arithmetic is axiomatic and our axioms of it are arbitrary and ad hoc. Because they're axioms. Who says aliens would come up with the same ones?

There are huge assumptions made in this thread that only indicate the brief time that the contributors have given to thinking about all this stuff. If you think about it for a couple of minutes, sure, it all feels very natural. Zero, infinity, division, mathematical logic, set theory, etc. But if you think about it a bit more, and then do a bit more than think, and go read about it, it's obvious that those are just the ideas that we chose to go with, not the only ones that exist, and certainly not the only ones proposed by mathematicians, logicians, computer scientists and philosophers over the years. For instance, Hilbert was a finitarian, division doesn't work with zero, zero doesn't work with division, dividing an infinity multiplies it, material implication leads to counter-intuitiveness paradoxes, set theory with only the membership relation leads to paradoxes, etc etc etc. Mathematics is full of unnatural holes that need constant patching up, and there is nothing to say that it is in any way, shape or form "real", let alone universal as so many people in this thread seem to be saying.

Aliens may have different biochemistry, but it would be made from the same chemical elements as ours. Likewise their formal systems may be wildly different from ours, but they will still be based on form (even implication is ultimately a very simple formal structure. Math doesn't even require causality as a prerequisite!)

Last but not least. many people (Kurt Gödel among them) believe that mathematical thought is actually perception of real phenomenon in a "higher" plane of reality, which, if true, seems to me to imply that alien mathematicians would be perceiving the same phenomenon as humans, literally. In this view, the "truths of mathematics" are literally the same "objects" for them and for us.

Almost certainly true. We as evolved creatures do not perceive objective reality, but only enough reality as to enable our ancestors to survive our very niche environment (niche relative to the entirety of the universe). Our science and mathematics model only our perceptions of reality, and not reality itself.

It is folly to assume that an alien, evolved along an entirely different set of initial conditions, would share our perceptions of reality. Our mathematics, modeling as it does our perceptions, serves human needs and perceptions only.

Can you give an example of a mathematical concept which could be different?

I believe that math is universal. We may use models to understand it (infinity, perfect circles, etc.), but the underlying mathematical truth is independent of humans. The same is true for science. There are physical laws which we look to discover. We use models in science to understand them, but the models are not the same as the underlying truth

The only thing that is "universal" is Nature itself. Mathematics, on the other hand, is a reflection of Nature in the human mind; or, put differently, it's Nature's language we humans are capable of understanding. It is therefore conceivable that other creatures, far removed from us, could "hear" a language that is just as far removed from ours.

Reality itself is beyond us. All that we have at our disposal are our perceptions, which is what we are modeling when we "do math". Mathematics is not universal, it's a language that we use to communicate what we perceive to other humans.

It's speculative, whether an alien species, with a perceptual and cognitive system evolved entirely elsewhere under other pressures, would have an understanding that intersects with our subset of reality. I personally think it's unlikely. What that would mean is that we wouldn't be able to communicate, never mind trade technology and mathematical ideas.

I mean, maybe, but this is conveniently unprovable, much like the flying spaghetti monster, since you are saying we could not communicate because our slices of reality don't intersect. I disagree and think that most likely we would live in a reality that was largely the same, but I guess we'll never know!

Perhaps. I was responding to the assertion that Mathematics exist independently of biology or perception of reality which is at least as unprovable

Where is this coming from? Is this relevant to what I wrote?

1) Math describes reality, so human math and alien math are the same thing, just different perspectives of reality and so, ultimately compatible.

2) Math is a language that humans use to describe their perceptions of reality to other humans. Perception is not reality, but a kind of isomorphism or pared-down heuristic, driven as it is by the evolutionary imperative to streamline for survival. Therefore a language that appears to be universal will only make sense to entities that employ the same perceptual framework for understanding, which is to say, human beings.

I agree that something that is "true" in human math will not be directly "false" in an alien math. 2+2 really does equal 4. The question is whether 2+2=4 is relevant or even understandable to an alien.

My point being, assumptions get baked-in to systems in surprising ways. Even something seemingly-objective like math. Especially when you're using it as the basis for communication, then what counts as "basic" or "fundamental" or "standard" reflects a perspective, not a fundamental truth.

It's likely their mathematical systems would eventually reach the same conclusions as ours, but the prominence or significance of fields or results (like circles and triangles) might be radically different. Even though we view those components as absolutely fundamental, it's possible an equivalent system could be built from different primitives.

But isn't the whole point to to do our best to bypass human-centric systems of understanding, and arrive at the "core truth" of the matter? Whether that's possible is another matter, but even if it's not possible, surely it's something that can be theoretically approached, and I would wager is precisely what PG means by "one mathematics."

> And then there’s the possibility that our mathematics and overall perception of reality is shaped by our biology in far deeper ways than we imagine and currently understand.

Yes, but also no. Consider some first principles:

-We have every reason to believe that any and all life would not live forever, or if the life in question is "intelligent" (a nebulous/human-centric term, for sure) would at the very least conceive of other things not lasting forever (such as stars, or even the universe itself [or, if you want to be really generous, "this current iteration of the universe"]). -Therefore we can reasonably assume that all "intelligent" life in the universe would understand the concept of scarcity (either via finite lifespans/time, food/energy sources, both, or something else), non-infinity. I'd go so far as to say that any life form that doesn't understand its own mortality or other such limits should be not be considered "intelligent," at least for the reasons of this discussion. -Therefore we can reasonably assume that said intelligent life would somehow conceptualize a binary state (you're either alive or you're not, you either have access to an energy source or you don't, etc), and consequently would somehow or another understand the concept of "zero," "nothing," etc, as well as its opposite, "something." And from there, would necessarily discern the differences between two states of "somethings" (the state of "something" that is "one" is different than the state of "something" that is "two").

I know I'm using a lot of loaded terms here -- "reasonably," "assume," "discerning" -- but just like we look for life by looking for the markers of life that we know were necessary for Earth (carbon, water, etc), we can look for intelligence that exhibits the properties that we understand it to have. We need some sort of frame of reference, after all, if we are to do anything other than simply flail. If that frame of reference is to be proven wrong, that's wonderful, but until that's the case, I don't think it's unreasonable to assume that the universe's "primitives" would be perceived in any truly, truly different way such that the species' interpretation would cause humans to rethink our own understanding of the universe's "primitives" from the ground up.

After all, conceiving a difference between hydrogen and helium requires being able to tell the difference between one and two (electrons, as well as separate elements themselves). And considering we have every reason to believe that those make up the majority of the mass in the universe, any "intelligent" life (there's that human-centric term again) can be expected to somehow conceptualize that difference, and thus, do something like counting, and thus, approach the same primitives of mathematics that we do. The approach might be different, but what they're approaching -- the very fabric of reality, hopefully as objectively as possible -- must be assumed to be the same (again, that is, until we're given compelling evidence to believe otherwise).

That said, I've never studied the philosophy of mathematics, so I could be talking out of my ass here, this is just the reasoning of a layman after all. If anyone reads this and goes "no you're way off base," I'd love to hear it!

With science it doesn't matter who does a given experiment, anyone else doing the same experiment will get the same results. There's no scope for disagreement about verifiable scientific facts. Just do the experiment and find out. If we forgot everything we know about science, in a thousand years if we rediscovered science, very quickly we'd rediscover all the exact same facts about the world again.

This is the key difference from religion, which has no such "falsifiability" equivalent.

The closest thing to "beliefs" is probably an individual following which of several competing theories is most likely correct -- but there's always the underlying basis that any of them might have evidence showing they're incorrect at any time, and one's view should adjust as a result.

Often this comes in the form of deferring to other people or a consensus view, which could be construed as "faith" but is different: If you asked me how the universe exists, I'd say the big bang theory is the best answer we have, but I don't understand enough about the underlying science to explain why nor can my brain comprehend the reality of it. I have no loyalty or allegiance to this view, though; I could be swayed to another theory if the big bang is ever proven false or if a better theory arises.

Further reading: https://www.theatlantic.com/science/archive/2015/11/why-scie...

https://blogs.scientificamerican.com/cross-check/was-philoso...

Presumably aliens wouldn't experience music the same way we do, especially if they don't have ears or physical bodies. But music itself might be interesting to them even if they don't really "get" it, due to the mathematical relationships involved. And they might have something analogous that we can't fully understand.

On the other hand it's also at least possible that aliens would understand our music perfectly well, and find it boring or atrociously bad due to our obliviousness to phase relationships, or the inherent out-of-tuneness of modern music, or how the tempo of the music is untethered from its pitch, or because our range of human hearing misses out on all the good stuff that happens above 15-20khz.

The beauty and importance of art and culture comes to mind, and so does humor and jokes. Try explaining, in scientific terms, what makes a joke funny, and the task turns out to be far harder than expected.

[1] https://en.wikipedia.org/wiki/Hex_(board_game)

[2] https://senseis.xmp.net/?TrompTaylorRules

"477. "If intelligent entities from other parts of the universe exist at similar or superior technological levels to ours, would they draw the same or at least similar philosophical conclusions to us?" That is a very good question. And my answer is this. They better f#%*ing do if they want to have any hope of withstanding our relentless, merciless onslaught."

But they don't often seem to engage with the overall point. In this case, I think that point is that there are probably local truths and universal truths, and wouldn't it be neat to focus on the universal ones, or figure out which ones are only locally true, and why? But who knows, that's just my interpretation.

I don’t think stable is the right word. A stool with three legs is less wobbly on an uneven surface.

A high stool with three legs is less stable to horizontal forces (forces towards the gap between legs).

I would say a tall stool with four legs is more stable (assuming the outer diameter of the footprint of the feet remains the same.)

It is the same reason three-wheeled cars are less stable than four wheeled cars.

If you want a stable seat on an uneven surface, you use a one legged stool (your legs make the other two adjustable legs).

Finally, the concept of a stool depends on two-legs and various other human specific environmental details. Aliens might not understand our stool analysis at all.

Answering these questions is significantly harder - impossible given today's science.

This essay reads like the ramblings of a man who's smoked too much weed.

Paul should stick to tech start ups - if you want serious insight around science, space and the universe hit up people like Richard Feynman...

https://www.youtube.com/watch?v=36GT2zI8lVA

This isn't even a shared principle among humans. How many experiments does it take for you to have 50% belief in a hypothesis?. What is the number of experiments? It's literally impossible to answer. It's not even clear what "belief" is or what 50% means.

This ambiguity of the word isn't even the main problem. If I run the same experiment with perfect observational tools 10 billion times and it verifies my hypothesis. Does that raise my belief further? What if on the 10 billionth and first time the test shows a negative result? That literally invalidates the hypothesis. Keep in mind we are assuming my observational tools are perfect. Does this make my belief shoot down to zero?

If this possibility of a negative result remains true after any number of tests then what does it say about belief? Why should I believe anything if a single negative experiment can invalidate 10 billion positive experiments (assuming perfect observational tools of course)?

Let me bring a more concrete example. I hypothesize all zebras have stripes. I observe zebras 10 billion times. They all confirm my hypothesis. Then on the 10 billionth and first time I see a zebra with spots. My hypothesis is wrong. This can happen any time.

Anyway to bring it back to his point. Don't assume shared axiomatic truths. PG already assumed that it's shared among humans. He's wrong. The nature of science and the scientific method is not universally shared or even fully understood among humans. He's likely also wrong about aliens as he is about humans.

Even if the original belief turns out to be wrong, you only have to slightly weaken it and it will remain true: "the vast majority of zebras have stripes". Even if you discover a new continent full of hordes of uniformly-colored zebras, the true hypothesis becomes "the vast majority of zebras in my original continent are striped".

Essentially every observation brings proof for a whole family of hypotheses. We normally only talk about the strongest of these hypotheses, but that doesn't meant that a negative example rules out the entire family.

For example, even if we didn't find a deductive proof the Fermat's last theorem was wrong even after all of the empirical proof that it probably wasn't, a weaker version would have still remained true - the one validated by that empirical proof.

The hypothesis does not remain true. It was never proven to be true and the new hypothesis is still not proven to be true. Science cannot prove anything to be true. I can find a cave full of of spotted zebras, and you have to further weaken your hypothesis of continents, I can then find that the stripes were actually microscopic spots and my perfect observation tool, though never wrong has limited resolution. Ad infinitum. Nobody ever considers your made up philosophy because it's changing the rules of the game. It's making a statement then adjusting your statement once it's proven wrong... people look down on that kind of thing.

What I'm writing here isn't something I pulled out of my ass. It's well known that in science, the scientific method, and reality itself, nothing can be proven. Proof is the domain of math and logic, not science. In science, things can only be falsified. To quote Einstein:

“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”

Einstein obviously isn't saying stuff like a single experiment causes me to adjust my hypothesis and divide split it into two different ones because it's kind of inconsistent.

There are people who truly understand science, but most of the population doesn't (including PG). I think what's going on with you is you're in the later camp, you've long held the incorrect belief that science can prove things and this long held ideology is coming into contact with the actual logic of the situation and your adjusting your belief to maintain a biased ideology.

Do you look up to PG? Bias can be corrected when an authority confirms the opposite. I quoted Einstein here. One of the ultimate authorities on science, a person who overturned the hypothesis about Newtonian physics being a model for motion. A single experiment proved it wrong and now Newtonian physics is simply an approximation that is ultimately wrong. Hopefully that will clear things up, if not... then you must be an Alien far more strange then what PG is describing.

That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.

Also, this is not unique to science. The same phenomenon can happen in mathematics or logic for theorems that have been neither proven nor disproven yet. We can perform numerical experiments to test a numerical theorem, and gain some amount of confidence in that theorem even if we haven't proven it to be true. We can often establish lower or upper bounds in the course of this experimentation, where we find that the theorem is True at least for some limited subset of all numbers - and this remains True and useful even if it later turns out that there exist counter-examples.

This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.

Please don't accuse me of acting condescending. It's very offensive and hurts my feelings when I'm accused of something I'm not doing.

I am criticizing you, but I am not being condescending. There is a huge difference.

Perhaps the alien thing was bad. I apologize for that. The intent was a joke and was not condescension.

>That is the point that I am trying to make: experimentation can bring proof to strengthen a hypothesis. Even if a later experiment invalidates a hypothesis, all of the previous experiments' results don't disappear, and any new hypothesis we formulate still needs to be coherent with them to have any value: we have actually learned something important from our thousand experiments, even if our 1001st showed that the hypothesis we had in mind was false.

Yes but this was not part of the discussion. We're talking about science as a principle. Not what we have learned from the process of science.

>This observation is also very important for understanding why the history of natural philosophy is essentially one of continuous progress, with very little backtracking: even if induction is not good enough to know that we have a perfectly complete and consistent theory (and we will never have one), we always have something salvageable from all of the experimentation done so far. Even geocentric models with their epicycles were actually working models, which predicted the positions of planets in the next 1000 years to quite good accuracy, even if they were clearly wrong in the end.

Important or not, we diverged from the point. Whether Science is a valid principle shared by humans and aliens is the point. My point is, PG's view isn't even shared with humans, why should he assume it's going to be shared with aliens?

You're talking about the importance of science. The value of science. That's off topic.

What we're discussing is "the principle that a controlled experiment testing some hypothesis entitles us to have proportionally increased belief in it". Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.

What PG was essentially talking about was that all humans agree on the value of inductive reasoning ("experience") *, despite the philosophical problem of induction. This is far older than any notion of science, and is universal among not just humans, but also life forms on Earth in general (at least plants, fungi, and animals). I honestly very much doubt that it is even possible to function in the world, let alone to build interplanetary communication, if you reject inductive reasoning.

The fact that we can't reconcile inductive and deductive reasoning is a limitation of our philosophical/logical/mathematical systems, not some ultimate truth that invalidates the above principle.

Quite off-topic, but I will also note that I don't really like PG in general, and think much of his argumentation style is unpleasant and often makes undue assumptions. I just happen to strongly agree with (my interpretation of) this particular point.

* admittedly, he did go a step further by mentioning "controlled experiments", which induction doesn't rely on; but I don't think that really modifies the statement. As a toddler, when you place a cube on the ground, look away, and then look back, you're performing a controlled experiment to check if objects have permanence.

It's not reconciling those two systems. It's reconciling all of logic (induction and deduction) and reality itself. Logic, by logic itself is inapplicable to reality. We live in a universe of unknown domains and imprecise/inconsistent measurements. At any point in time we can make an observation that contradicts a previous observation. This makes proof impossible. While proof is impossible because of the possibility of a contradictory observation, falsification is very possible. The goal of science is falsification, not proof.

This is the logical conclusion of science and therefore reality. Logic, deduction and induction and proof are mostly the domain of mathematics or little axiomatic games we play where we artificially limit the domain. It's a Very very different domain from the one science operates in.

Most of humanity actually agrees deduction and induction are inapplicable to reality. Hence why science is, in the end, the most rigorous form of determining truth (despite the fact that it actually can't) instead of logic. This is in fact the conclusion reached ABOUT reality when we apply logic to it; that logic itself is inapplicable to reality as we know it.

When we check if a cube on the ground 100 times and see that it exists but we can't know what the next 10 billion observations will yield. Perhaps the 100 observations were biased, and the 10 billion subsequent observations yield that cube was a reflection, the toddler was mistaken and the situation did not exist long enough for the toddler to observe the cube past 100 observations.

> Anyone that rejects this principle doesn't know if the sun will rise up tomorrow, is terrified that they may fall through the floor at any moment, or worse, drift off into the enormity of space.

This is my issue with PG. If you look at science rigorously... we actually don't assume this is true. Science cannot verify whether the sun will rise tomorrow or whether or not we will or will not fall through the floor. That is science in a nut shell. PG is saying something WRONG about science and that Aliens will share a belief with us about it.

As for our day to day experiences, you're right. We all believe the sun will rise tomorrow, but this isn't science. This is simply bias, that all humans are born with. We ASSUME the sun will rise tomorrow, but there is no form of reasoning (scientific, deductive or inductive) that can lead us to that conclusion. PG was NOT talking about this. He was talking about Science and controlled experiments. Not shared assumptions about reality.

If PG said, "We assume that Aliens, like us, assume that when we're not looking the cube still exists even though we only observed it a couple of times." then I can probably get behind that, but it is an entirely different statement.

So then you agree that "experimentation can bring proof to strengthen a hypothesis"

> Whether Science is a valid principle shared by humans and aliens

Well, the experimentation part, that can bring proof to strength a hypothesis, is something that you agreed to.

So that part would be shared, that you agreed to.

>Well, the experimentation part, that can bring proof to strength a hypothesis, is something that you agreed to. >So that part would be shared, that you agreed to.

Never agreed. You misinterpreted. I agreed to this: "we have actually learned something important from our thousand experiments". You learned that for 1000 experiments you observed something. That's it.

Ok great, so then you agree that this is a principle that would be shared, which is the point.

So then yes, that is an agreement that at least to that statement.

> But established nothing.

Well it established that at least you agree with that statement.

> Because I disagree with you.

actually you said this "I agreed to this:"

This is you saying that you agreed with the statement that followed that. So yes, you used the word agree.

Ok, you agree with this then.

Yes, that is my point. This is what you agreed with.

You are agreeing that at least something has been learned.

That is a valuable point to make, in and of itself, that at least something has been learned.

Valuable,.depends on your opinion, useless to proving a point yes.

https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_induc...

So why would aliens hold this "principle" the same if humans don't even agree on it? PG is wrong. His own principles upended not even by aliens, but by humanity, thus how accurate can his assumptions about universal principles even be? No that accurate imho.

There is no divide, there is the illusion of divide because we didn't have a rigourous formal model of how to build reliable knowledge and everyone focused on different but relevant aspects.

Bayesian reasoning is the correct way if you have justifiable priors, but we didn't have a way to calculate the correct prior.

Solomonoff showed us how with his theory: Kolmogorov complexity is a measure of parsimony, and this is how to select priors in a formal, rigourous way.

Solomonoff induction is to knowledge what Turing machines or the lambda calculus are to computation. Sure, aliens might not discover Turing machines exactly, or the lambda calculus exactly, but whatever they do build that's capable of universal computation, we already know it must be isomorphic to a Turing machine, because all constructions capable of computation must be by necessity.

The frequentist/Bayesian divide is a separate issue about how to interpret statistical data in useful ways, not specifically about how we know what we know and what confidence we should have in our knowledge, which is what you were asking about.

https://www.lesswrong.com/posts/Kyc5dFDzBg4WccrbK/an-intuiti...

I already disagree here. Simplicity is a heuristic in the brain. Like all heuristics it helps you solve some problems easier, but behind the word stands the entire life experience of someone, and whatever is "simpler" depends on way too many human things in his life. "Simple" is something human. So is Occam's razor.

Minor nitpick, but the subfield of Epistemology (or Alien Epistemology) might be a better term. Literally the study of knowledge, or how we know what we know and what is true.

https://plato.stanford.edu/entries/epistemology/

But trying to look at it from an alien (or AI) point of view is a really interesting thought experiment. What are universal truths in the literal sense of the word "universal"? And what are merely helio-centric cultural trends masquerading as universal truths.

I wonder if that's ever been done before. There's a little bit of discussion of it online, but not much it seems:

https://www.google.com/search?q=%22alien%20epistemology

Someone (typically at Bell Labs in like 1973) has nearly always already had my thought, but then explored it in much richer, finer detail than I have. It's disappointing to learn I'm not an unmatched genius, but much more illuminating to read and think about how others have considered similar issues.

I don't fault Paul for not doing this, it happens to everyone, and I imagine he's quite busy. I do fault all the people who read this draft and didn't do this very easy exercise for him. I think it reflects a undue deference towards frankly uninformed and pretty basic observations.

vs

>I wouldn't want to bet that all intelligent beings would understand the concept of justice

It would be unjust if a technologically advanced society murdered all its scientists for heresy during a religious coup, but Paul is unwilling to bet that all alien societies understand justice. Why the belief in science then? Our own planet has theocracies and a President of the supposed #1 country has updated scientific hurricane reports with a sharpie.

Plenty of science fiction authors have speculated meeting theocratic aliens. Peter Hamilton's Salvation comes to mind (you get where that series is going from the title).

I suspect they will even recognize Go's 19x19 board (b/c it has properties other board sizes would not have).

But would they recognize 12 tones scales?

It's possible that they might, but if they have any kind of musical discernment they might find it objectionable unless they're as lazy as we are.

12-tone equal temperament works more-or-less because it gives good-enough approximations of almost all the small-number ratios that a musician making relatively straightforward music would care about. However, thirds and sixths are pretty out-of-tune and frequency ratios with sevens in them like 7:6 and 7:4 don't have any reasonable approximation.

On the other hand, 41-tone equal temperament is a much better system in just about any way you might care about, except that it makes building physical instruments more complicated because there's more notes to worry about. 53 is also really good, but that's even more notes.

And then there's pure just intonation, which is a great option if you don't really care about being able to transpose freely to/from any key, but you want each note pitch to be dead-on exact.

In other words, 12-TET is a kind of local optimum that we're stuck in for historical reasons and because the barriers to change are high. You might as well ask if an alien civilization would recognize the concept of daylight-savings-time, or the QWERTY keyboard, or the electoral college.

Assuming they have musical instruments and don't have far more digits (fingers or other dextrous appendages) than us, they probably would, and even if they did have more digits, it'd seem like a fairly obvious choice for a species with our anatomy. Especially for equal temperament, but also for just intonation, a 12-tone scale provides exceptionally good approximations or numerous instances of the simplest possible frequency ratios, so any creature that has to limit the number of tones on their instruments for practical considerations will have a high likelihood of stumbling on a 12-tone scale even before developing an understanding of why it actually works so well and/or find it a very compelling option given a more in-depth analysis.

Relatively? More likely. "Oh yeah, we get that. At a number of levels which we call honktaves, incidentally."

Absolutely? As in, we have that set of frequencies bolted on to our culture too? Seems less likely.

More at: https://corinth.kardianos.com/

I think that one depends a lot on how they learn or can transfer knowledge. We rely on language for it, and language is severely limited - We can't learn olympic gymnastics from watching TV, but a species that could directly transfer memories and behaviors would have a huge leg on us in that regard - I'd assume we'd quickly notice their fast-paced technological advance.

Or completely miss it, because they'd anyhilate themselves a couple hours after discovering the military use of nuclear fusion ;-)

Apparently the feed was created by Aaron Swartz himself, so I wouldn't think there would be any errors there. But it looks like it's hosted via Aaron's own site, so perhaps there's some problem caused by it not being maintained?

I think this has long happened, and do not understand why this is still presented as "truth".

I prefer the standard of truth used by natural-sciences. The rest I find pretty bendable (virology included).

It doesn't though. Firstly, Occam's razor is not about simplicity but about "parsimony". Parsimony is calculated in information theory via Kolmogorov complexity:

https://en.wikipedia.org/wiki/Kolmogorov_complexity

This then lets us describe how to do induction so it provably converges in the fastest way possible, Solomonoff Induction:

https://en.wikipedia.org/wiki/Solomonoff%27s_theory_of_induc...

Arguably, this can be seen as a formalization of the scientific process.

In theory perhaps, but in practice it is firstly calculated by the mind of the person who plays the Occam's Razor card, and then subsequently by people who ingest the claim, and typically all participants are performing their calculations using biased heuristics and flawed logic, and have negative interest in what is actually true, or if truth is even reachable.

I was thinking more "triangular sandwiches taste better than square ones", but you do you!

The really esoteric branches of mathematics, I could see huge variances. Materials sciences would vary widely. But the basic high school stuff? I'd lay odds on it.

I have no idea how one would go about proving this but personally I would start with something like the "Principle of Least Action" applied to thought.

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being by George Lakoff

https://www.goodreads.com/book/show/53337

Wow some people still think we're alone. Wow. Just, wow.

The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.

And likewise, would we be able to understand a non-human proof?

I'm a puzzled by the confidence here. I would assume pg is at least minimally familiar with some of the key philosophical themes and schools of thought around this topic.

> For intuitionistic mathematics ... truth is a mental construct of a proof in a language, and only shared by communicating that construct to other minds.

Even assuming you are fully and accurately representing the intuitionist view, you must be aware that there are competing schools of thought with strong pedigrees, like mathematical platonism, that are grounded in a more realist view of mathematical objects.

PG didn't go so far as to stake out that position here in the essay, but his thought experiment leverages a view of mathematical truth that hues closer to this (platonist) camp.

> The question is whether we can communicate our mathematics to non-humans ...

That is an interesting question, but pg did not ask that question in the essay, and its answer doesn't seem relevant to the point he was trying to make.

I think pg is educated, at least the broad outline, of the age-old controversies in this topic and I'm grateful as a reader that he spares me the details and assumes some preliminary context.

His style of pop philosophy is useful in startups because it does not require context. It is ungrounded in actual philosophy, and would struggle to find an audience outside of this group.

But it is not straightforward to me. Or, in my impression, most philosophers. I'm reluctant to agree that aliens will assign any importance or even understand concepts like geometry or calculus. So to me, it is not preliminary context but it is a critical part of the discussion.

But even a brief foray into discussions by actual philosophers will show that to not be true. What Paul Graham is doing is providing small subcultural insights tailored to an audience with a passing interest in philosophy.

I think the word in that idiom is 'hews', which means to adhere strictly to a standard, probably from the sense of the word meaning to strike or cut or beat - often used to talk about cutting a tree into shape.

I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.

Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.

How can we communicate actual axioms to extra-terrestrials?

Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.

His very first premise "The truths of mathematics would be the same, because they're true by definition" shows an ignorance of an entire field of discourse called the philosophy of mathematics. Whether aliens would have the same math as us is one of the fundamental questions, and to presume the answer shows a disdain of the existing body of thought on this topic.

In particular, we humans still hotly debate which math is the "right" math. Is the axiom of choice "true"? Does it make sense to say that it has a truth value? Is such a truth value amenable to human discovery? Does PG know the answers to all of these questions?

PG is either unaware of the human-choice nature of the fundaments of math or he thinks he has special knowledge of it. Either way makes this essay one of my least favorites.

Whether or not it's possible to prove a specific postulate in set theory is formally true (the axiom of choice, which was only formulated in 1904) isn't going to stop aliens counting objects and calculating the area of squares. They may not agree with us about formalism vs intuitionism, but it seems likely they would agree with us about a huge array of practical mathematical operations and results. PG is just asking what are the areas we would be extremely likely to agree on. Pointing out that there might be specific, advanced, highly obscure controversies we might disagree on isn't even in contention. Of course there are.

For example, what if aliens are so large that the non-euclidian nature of space becomes obvious to them? Perhaps they would never come up with euclidian geometry. Maybe they would never create the notion of pi. Yes it shows up in other areas but you can certainly imagine an alien civilization that has a different circumference-to-radius ratio or disavows the entire idea of having a constant circumference-to-radius ratio.

Newton's laws of motion strictly speaking are not true, Einstein proved this, but they are incredibly useful as is the mathematics we use to formalise them. We know that the quantum mechanics and relativity theories we have now and not complete, but that's beside the point.

PG is simply pointing out that since aliens, if they exist, live in the same real world we do they will discover a lot of the same practical results we do, and then goes further to suggest they might agree with us about a lot of the less strictly provable ones as well. Going "Oh well but this obscure question here would still be in contention" is, well, somewhat sailing past the point by a little smidgenette. It's criticising a claim I don't think it's reasonable to think PG is making.

I believe our difference is that you are presupposing that aliens will have essentially the same experience with reality that we do. I agree that if somehow a separate human civilization evolved in another galaxy, their math would likely look similar to ours. But you can imagine aliens that have very different experiences of "what actually works in the real world" (due to experiencing a different set of physics, such as at a different physical scale) and thus come up with different math.

There's a character Rimmer in a comedy series Red Dwarf that's always going on about how aliens would be unimaginable to use because they're alien and everything they do would be... well.. 'alien'. It's poking fun at SF that presents aliens as inherently utterly unintelligible to us simply by virtue of being alien, but frankly that's absurd. Physics doesn't care who or where you are, the same rules apply.

It's not your brain's fault, really. It doesn't know any better. Your brain has a way of interpreting the sensory perceptions communicated to it. It conflates this with 'reality' because it doesn't know anything thing else. And because it doesn't have anything else, it also declares itself the universal understander. Just like there are sounds you cannot hear, and chemicals you cannot taste, wavelengths you cannot see -- consider that there may be thoughts that you physically cannot think.

No magic and no spooky mysterious stuff. Just that it seems quite possible that our brains are limited in their ability to construct a representation of the universe, and insofar as the representation is true, we have kno way of knowing if it's the only true one.

Seriously, there are sensory experiences I can’t have, but in principle no thoughts that are intractable to analysis and understanding given enough information.

It seems pretty obvious that there are thoughts that are too big to fit inside our heads.

Pointing out possible edge cases is a non sequitur. Of course there will be things we don’t have in common. Yes, we know that. The fact we’re explicitly looking at common ground from first principles implicitly acknowledges there will be ground not in common. That’s given.

As a limited computing system, our brain has been optimized to process the world in specific ways using various heuristics and compression techniques. Alien brains would presumably be optimized in different ways. Given our limited capabilities, it seems quite possible that translating any concepts between the two systems would be unfeasible for either our brain or the alien one.

I would like to think that there would common ground but that isn't something that can be known a priori, and is thus not presently "proveable".

Edit: We do have some fairly "alien" brains here on earth but we haven't had much luck teaching any mollusks any of our mathematical truths.

I see it as being on the third floor, but that's only my brain lying to me, right?

That could be possible, but I don't think that we can assume that.